I was watching this video about the HK416 and at about 37 seconds into the video, the demonstrator says that the weapon has a free-floating fore-end. I also saw on the Knight's Armament company's webpage, some of the products are described as having a free-floating barrel. What does this mean?

It generally means that the barrel doesn't touch the fore-end at all. The fore-
end is attached to the receiver below the barrel but does not support the barrel.
The benefit of a free-floated barrel is that the fore-end cannot apply
pressure on the barrel at any point, and the barrel can dissipate heat evenly.
This minimizes barrel warp and improves accuracy.

Think of it in this way. If you have your rifle in shooting position, with your hand holding up the barrel by the fore-grip, then you are pushing up on the barrel by the force necessary to oppose gravity. This pressure can affect accuracy by distorting the barrel's angle. If you put your gun into a theoretical vise at the receiver, and pushed on the barrel with varying pressure, you could affect where the bullet hit on the target. A free floating fore-grip removes this variable, whether it is on a bolt action gun stock, as shown above, or an AR-15 with a separate grip.

Whether it matters depends on the situation, the gun, and how far you're shooting.

Note: As a bullet traverses a barrel, the barrel twists and whips minutely. Although we cannot see this movement with the naked eye, any contact with the stock as it makes these contortions can adversely affect accuracy. Hence the free float usually helps consistent accuracy